In mathematics, a telescoping series is a series whose general term ${\displaystyle t_{n}}$ is of the form ${\displaystyle t_{n}=a_{n+1}-a_{n}}$, i.e. the difference of two consecutive terms of a sequence ${\displaystyle (a_{n})}$. As a consequence the partial sums of the series only consists of two terms of ${\displaystyle (a_{n})}$ after cancellation.

The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences.

An early statement of the formula for the sum or partial sums of a telescoping series can be found in a 1644 work by Evangelista Torricelli, De dimensione parabolae.

Telescoping sums are finite sums in which pairs of consecutive terms partly cancel each other, leaving only parts of the initial and final terms. Let ${\displaystyle a_{n}}$ be the elements of a sequence of numbers. Then $${\displaystyle \sum _{n=1}^{N}\left(a_{n}-a_{n-1}\right)=a_{N}-a_{0}.}$$ If ${\displaystyle a_{n}}$ converges to a limit ${\displaystyle L}$, the telescoping series gives: $${\displaystyle \sum _{n=1}^{\infty }\left(a_{n}-a_{n-1}\right)=L-a_{0}.}$$

Every series is a telescoping series of its own partial sums.

In probability theory, a Poisson process is a stochastic process of which the simplest case involves "occurrences" at random times, the waiting time until the next occurrence having a memoryless exponential distribution, and the number of "occurrences" in any time interval having a Poisson distribution whose expected value is proportional to the length of the time interval. Let X_t be the number of "occurrences" before time t, and let T_x be the waiting time until the xth "occurrence". We seek the probability density function of the random variable T_x. We use the probability mass function for the Poisson distribution, which tells us that

where λ is the average number of occurrences in any time interval of length 1. Observe that the event {X_t ≥ x} is the same as the event {T_x ≤ t}, and thus they have the same probability. Intuitively, if something occurs at least ${\displaystyle x}$ times before time ${\displaystyle t}$, we have to wait at most ${\displaystyle t}$ for the ${\displaystyle xth}$ occurrence. The density function we seek is therefore

The sum telescopes, leaving